Arrow’s Impossibility Theorem.
Any voting method that uses voters’ ranked choices to decide an election with a single winner will violate at least one of the four criteria we’ve discussed: majority, monotonicity, Condorcet, or IIA.
Section 7.4 Arrow’s Theorem, Conclusions and Exercises
So Where’s the Fair Method?
Here is a summary of whether each of our voting methods satisfies each of our fairness criteria:
| Majority Criterion | Monotonicity Criterion | Condorcet Criterion | IIA Criterion | |
| Plurality Method | Satisfies | Satisfies | Violates | Violates |
| Borda Count Method | Violates | Satisfies | Violates | Violates |
| Instant Runoff Voting (IRV) Method | Satisfies | Violates | Violates | Violates |
| Pairwise Comparison Method | Satisfies | Satisfies | Satisfies | Violates |
So far none of the voting methods we have studied satisfy our four fairness criteria. At this point, you are probably asking why we keep looking at method after method just to point out that they are not fully fair. We must be holding out on the perfect method, right? Unfortunately, no.
To see a very simple example of how difficult voting can be, consider the election below:
| Number of votes | 5 | 5 | 5 |
| 1st choice | A | C | B |
| 2nd choice | B | A | C |
| 3rd choice | C | B | A |
Notice that in this election:
| 10 people prefer A over B |
| 10 people prefer B over C |
| 10 people prefer C over A |
No matter whom we choose as the winner, 2/3 of voters would prefer someone else! This scenario is dubbed Condorcet’s Voting Paradox, and demonstrates how voting preferences are not transitive (just because A is preferred over B, and B over C, does not mean A is preferred over C). It also demonstrates the impossibility of identifying a winner this election -- because there is no fair resolution.
Not only is it true that none of our voting methods satisfy all of the fairness criteria we care about, even worse, it turns out that there is no perfect voting method.
Subsection 7.4.1 Arrow’s Impossibility Theorem
In 1949, a mathematical economist named Kenneth Arrow was able to prove that there is no voting method that will satisfy all the fairness criteria we have discussed.
In other words, Arrow’s theorem means that no matter how hard we try to find one, we cannot find a voting method using ranked choices to find a winner that satisfies every fairness criteria we have studied here. Consequently, in any election a losing candidate can claim the outcome is unfair in some way… and be correct. This puts democracies in a difficult position: how can a winner of an election be determined in a manner that all constituents believe is consistently fair?
The answer is, it can’t. Instead, we change the question and focus on finding a voting method that satisfies the most – or the most important – fairness criteria. Interestingly, the fact that there is no single fairest method might explain why there are many different voting methods used throughout the world.
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commons.wikimedia.org/wiki/File:Electoral_systems_map.svgThe map in Figure 7.4.4 shows the systems used to elect the lower house of each country’s legislature. You can see a larger version if you right-click and select “view image”. Note that the Borda count is included in the “other systems” category (green), but is used in only one of those countries: the Pacific island nation of Nauru. It is also used, however, to elect two seats in the parliament of Slovenia that are reserved for the Hungarian and Italian ethnic minorities.
There are so many different voting methods, including ones we haven’t discussed in this chapter, and all have their pros and cons. Fairness criteria can’t tell us which method is the best, but they can at least provide some useful information to consider. It is then up to a given community to consider the many options thoughtfully and chose one that will help it make the best collective decisions that it can.
Subsection 7.4.2 Chapter 7 Exercises
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Use the results of Election 1 as shown in the table to answer the questions below.
Table 7.4.5. Preference Table for Election 1 Number of votes 26 25 15 34 1st A B B C 2nd C A C A 3rd B C A B - How many total voters were there?
- True or False: a total of 26 voters preferred Candidate A over Candidate B.
- Who is the plurality winner? Does that candidate have a majority?
- Calculate the winner of Election 1 using the Borda count.
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The table below shows presidential election results for Florida in 2000, between George W. Bush, Al Gore, and Green Party candidate Ralph Nader. The vote totals, given as percentages, are the real results.The voters’ second and third choices are educated guesses based on what we know of voters’ opinions during that election.
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Simplified to exclude candidates with less than 1% of the vote.Table 7.4.6. Preference Table for 2000 U.S. Presidential Election in Florida Percent of votes 49.2 49.1 1.7 1st Bush Gore Nader 2nd Gore Nader Gore 3rd Nader Bush Bush - In this example, does a Condorcet candidate exist? If so, who is it?
- What (if anything) does this election tell us about the plurality method and the Condorcet criterion?
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Calculate the winner of Election 2 (shown below) using IRV.
Table 7.4.7. Preference Table for Election 2 Number of votes 17 15 5 10 2 1 1st A B C C D D 2nd D C A D C B 3rd B A D B B A 4th C D B A A C -
The table below shows the ballots for Election 3. It is the same as Election 2, except that some voters changed their first place votes from A to C.
Table 7.4.8. Preference Table for Election 3 Number of votes 5 12 15 5 10 2 1 1st C A B C C D D 2nd A D C A D C B 3rd D B A D B B A 4th B C D B A A C - Calculate the winner of Election 3 using IRV.
- Compare Election 2 and Election 3. Do these elections show IRV violating one of the fairness criteria? If so, which one, and why?
